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What you're watching here is a torus cycling through its eigenfunctions...the surface's pure standing-wave modes. An eigenfunction is a mode that, when the surface Laplacian operator acts on it, reproduces itself scaled by a constant...that constant fixes the natural frequency. Each mode is indexed by two counts: how many... show more
54,693 views • 11 months ago •via X (Twitter)
33 Comments
A cone for Gandalf 😁
Wondering if the standing waves can evolve over time moving over the surface of the torus, such that it looks like flowers blooming:
my work is so inferior it is perturbed....
Looks like a good start...Im sure it will get back with iterative improvements
hahaha thanks! workin on it
Had to try this out using my discrete simulation framework - here is a standing wave wrapped around the torus in different winding directions. As the mode shifts, the surface breathes, twists, and ripples along its normal direction, revealing how geometry stores and exchanges energy across different harmonic shapes.
Never mind...figured it out...buying you a coffee for the inspiration, thank you !
Eventually it will take on the shape of an egg.
I will run it for a longer time and see
Give us layman applications that change the average life today.
Field coherence. Eignestate/Hamiltonian - resonance. 🌊
Pineal Gland... Pine Cone... Interesting... @grok any thoughts about this post and that intuition?
….if only people could see the poetry in this…. 🎶❤️
This animation reminds me of a "UFO" I once witnessed. I would describe it like this: A tightly packed cluster of ovoids, with increasingly bigger eggs towards the center, arranged in a radially symmetric pattern. Evocative of a lotus flower. 🪷 #hyperdonut
Astonishing!! Beautiful!
Absolutely beautiful
but actually don't stop i love it
@grok this resembles a lot of the electronic orbitals, don't you think?
Hedgehogging your torus Interesting that its just gaussian domains. You'd think there might be a hexagonal tiling too
Thousand Petalled Lotus
Looks like atomic orbitals.
Yeah curvature can be complex. Important to remember that it is all defined by angular force versus vibrational force the basis of our universe.
Unclear The sound of it all that continues to make sense into non sense is grating
Ummm..yes. And that’s super cool viz but you need to have it play the associated hz instead of that music
so... interuptions in space time creating tunelling activity motivated by symmetry breaking would complicate this model into the structures of remaining matter and resonances between geode and amplituhedron as duals of a sort?
Wow, koja količina kreativnosti! Ovo je sjajna Grok Imaginacija. Fantazija! 🌟👏👏👏
🐾🎨
Facts, as experimentation proves.
Interesting
Excellent visualization
How beautiful!
Genial 😍
This looks like the lattice I am have built with grok. It incorporates all the known lattices.
![String Theory Lecture 1 A String Does Not Move Like a Point A point particle traces a line through spacetime. A string traces a surface. This is the first geometric shift in String Theory. Particle mechanics asks where one object is at time t, so its history is a curve. String Theory asks where every point of an extended object is at worldsheet time τ, so we need another coordinate telling us where we are along the string. For a point particle x(t) So, for one input of time we get a position in Spacetime. For a string Xᵘ(τ,σ) Here τ plays the role of time on the worldsheet, while σ labels position along the string. Freeze τ and vary σ, and you see the string at one instant. Let τ move, and that curve sweeps out a two-dimensional surface... the worldsheet. The same comparison appears in the action. For a relativistic point particle, the geometric action measures worldline length S = −m ∫ ds If we parameterize the path by t, the action has one integral, one parameter, and one tangent vector dxᵘ/dt For a string, the same idea grows by one dimension. The action measures area, not length. In Nambu-Goto form, S = −T ∫ dτ dσ √[−det hₐᵦ] Here T is the string tension. It plays a role similar to mass, but for an extended object. It weights the area of a surface rather than the length of a line. The particle action has ∫ dt because the history is one-dimensional. The string action has ∫ dτ dσ because the history is two-dimensional. We are no longer summing along a path, we are summing over a surface. The geometry changes for the same reason. For the particle, one derivative is enough dxᵘ/dt For the string, the geometry is built from two derivatives: ∂τXᵘ and ∂σXᵘ The first tells you how the string changes as worldsheet time flows. The second tells you how the embedding changes as you move along the string. Together they define the induced worldsheet metric hₐᵦ = ∂ₐXᵘ ∂ᵦXᵤ In plain terms, hₐᵦ measures tangent lengths and tangent angles on the worldsheet. From it, the area element is dA = dτ dσ √[−det hₐᵦ] This, the Nambu-Goto action is the direct analogue of the point-particle length action. The point particle extremizes length and the string extremizes area. For calculations, people usually switch to the Polyakov action: S = −(T/2) ∫ dτ dσ √[−γ] γᵃᵇ ∂ₐXᵘ ∂ᵦXᵤ This describes the same classical string dynamics, but the algebra is cleaner. After choosing conformal gauge, varying with respect to Xᵘ gives (∂²/∂τ² − ∂²/∂σ²) Xᵘ = 0 This is the first real dynamical payoff... a two-dimensional wave equation on the worldsheet. For a point particle, the equation of motion tells you how one position evolves along one path. For a string, it tells you how an entire curve evolves, with waves traveling along it. The term ∂²Xᵘ/∂τ² measures acceleration in worldsheet time, while ∂²Xᵘ/∂σ² measures curvature along the string. The time evolution is balanced by how the string bends along its own length. This is why strings have oscillation modes. A point particle has one trajectory. A string has many possible vibration patterns, each one a normal mode of the worldsheet wave equation. For a closed string, σ wraps around the loop Xᵘ(τ, σ + 2π) = Xᵘ(τ, σ) For an open string, one standard free-end condition is ∂σXᵘ = 0 at the endpoints. Solving the wave equation gives waves moving in opposite directions along the string Xᵘ(τ,σ) = Fᵘ(τ + σ) + Gᵘ(τ − σ) A function of τ + σ moves one way. A function of τ − σ moves the other. Therefore, a particle has a worldline, its action measures length, and its geometry uses one tangent. The string has a worldsheet, its action measures area, and its geometry uses two tangent directions. #StringTheory #TheoreticalPhysics #MathematicalPhysics #Physics #Spacetime](https://image.24vids.com/tw-2047925796963004514/amplify_video_thumb/2047925638485753856/img/d0IV19m4OAIfEf0I.jpg)